Is there a permanent hacking strength bonus after a successful hack?

Solution 1:

Not After Every One

Successfully hacking something will not automatically increase your specialists' hacking scores.

Just When It's the Reward

When you are hacking an objective, the listed reward will sometimes be "Enemy Protocol". This is a permanent boost of 20 to the soldier's hack stat. You need to select this as your reward and succeed in order to get the boost.

From my own experience, I can say that the boost is applied immediately. Any further hacks the same mission will be easier.

This is mentioned as one of the ways to increase a soldier's hacking stat in Chris Hayes' answer to How can I raise a soldiers hacking ability?

Reports Otherwise Were Untrue

I've found some reports of getting a small increase in hacking skill after every successful hack, but, as discussed in this Reddit thread, that was likely confusion from the increase in hacking your specialists get every time they are promoted.

In that thread, multiple people report testing this idea, and finding that successfully hacking did not automatically increase hacking skill. One even inspected the source code. Chris Hayes has also checked the source for this bonus, and says that "I've searched repeatedly and can't find anything at all to support it."

Solution 2:

Your soldiers do not usually get a permanent bonus to hacking strength after a successful hack.

There is, however, a hacking reward called "Enemy Protocol" that you can win, which does grant the hacker a permanent +20 point increase to hacking, but it's a randomly generated reward, and, well, you have to successfully win it.